Pear Curve

Author

Eric W. Weisstein
February 20, 2012

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/PearCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PearCurve.html.

©2012 Wolfram Research, Inc. except for portions noted otherwise

Equation

Entity["PlaneCurve","PearCurve"]["CartesianEquation"][a][x,y]

With[{a=2},RegionPlot[(a^2)>(x^2)+2 (x^3)+5 (x^4)+6 (x^5)+6 (x^6)+4 (x^7)+(x^8)+(y^2)+2 x (y^2)+2 (x^2) (y^2)+4 (x^3) (y^2)+14 (x^4) (y^2)+12 (x^5) (y^2)+4 (x^6) (y^2)-3 (y^4)-2 x (y^4)+10 (x^2) (y^4)+12 (x^3) (y^4)+6 (x^4) (y^4)+2 (y^6)+4 x (y^6)+4 (x^2) (y^6)+(y^8),{x,-2.2,1},{y,-1.5,1.5},AspectRatio->Automatic]]

pear[a_]:=Factor[ComplexExpand[#Conjugate[#]&[Nest[x+I y+#^2&,0,3]]]]-a^2

Expand[pear[a]]

eqn=-(a^2)+(x^2)+2 (x^3)+5 (x^4)+6 (x^5)+6 (x^6)+4 (x^7)+(x^8)+(y^2)+2 x (y^2)+2 (x^2) (y^2)+4 (x^3) (y^2)+14 (x^4) (y^2)+12 (x^5) (y^2)+4 (x^6) (y^2)-3 (y^4)-2 x (y^4)+10 (x^2) (y^4)+12 (x^3) (y^4)+6 (x^4) (y^4)+2 (y^6)+4 x (y^6)+4 (x^2) (y^6)+(y^8);

AlgebraicDegree[pear[a],{x,y}]

Plot

Level Curves

<<Graphics`ImplicitPlot`
<<Utilities`Typesetting`

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
{
ImplicitPlot[pear==4,{x,-5,5},Ticks->None,AspectRatio->Automatic,
PlotLabel->TraditionalForm[r==2],PlotStyle->Red],
Show[ImplicitPlot[pear==#,{x,-5,5},Ticks->None,AspectRatio->Automatic,PlotStyle->Hue[(1-#)/5]]&/@Range[.5,4,.5]
]
}]
]]

Contour Plot

DensityPlot[pear,{x,-2,1},{y,-1.2,1.2},Mesh->False,ColorFunction->(Hue[2#]&),PlotPoints->100]

ContourPlot[pear,{x,-2,1},{y,-1.2,1.2},ColorFunction->(Hue[#]&),ContourLines->False,Contours->20,PlotPoints->100]

V6

ContourPlot[pear[#]==0,{x,-2.2,1},{y,-1.5,1.5},AspectRatio->Automatic]&/@{.3,.5, 1,2, 3,4}

Properties

Area

<<Graphics`InequalityGraphics`

InequalityPlot[pear<4,{x},{y}]

<<Calculus`Integration`

NIntegrate[Boole[pear<4],{x},{y}]

(area=With[{a=2},Integrate[Boole[(a^2)>(x^2)+2 (x^3)+5 (x^4)+6 (x^5)+6 (x^6)+4 (x^7)+(x^8)+(y^2)+2 x (y^2)+2 (x^2) (y^2)+4 (x^3) (y^2)+14 (x^4) (y^2)+12 (x^5) (y^2)+4 (x^6) (y^2)-3 (y^4)-2 x (y^4)+10 (x^2) (y^4)+12 (x^3) (y^4)+6 (x^4) (y^4)+2 (y^6)+4 x (y^6)+4 (x^2) (y^6)+(y^8)],{x,-∞,∞},{y,-∞,∞}]])//Timing

Integrate[Boole[pear<4],{y},{x}]

Implicit Curvature

FullSimplify[ImplicitCurvature[pear[a],{x,y}]]

Inflection Points

κ=((((1+x (((1+x))^2)))^2)+(-3+x (-2+x (8+x (8+3 x)))) (y^2)+(2+x (4+3 x)) (y^4)+(y^6)) ((((1+2 x (1+x) (1+2 x)))^2) (1+2 x (1+x (1+x (2+x (7+2 x (3+x))))))+2 (-5+2 x (-17+x (-83+x (-166+x (-77+4 x (78+x (230+3 x (108+x (87+8 x (5+x)))))))))) (y^2)+2 (9+2 x (-13+x (-91+12 x (6+x (71+2 x (69+x (69+10 x (4+x)))))))) (y^4)+4 (-19+8 x (1+x (43+x (114+x (153+40 x (3+x)))))) (y^6)+8 (31+6 x (15+x (33+20 x (2+x)))) (y^8)+48 (3+8 x (1+x)) (y^10)+64 (y^12));

Solve[eqn==κ==0,{x,y}]

InflectionPoints[eqn,{x,y}]

Singular Points

SingularPoints[eqn,{x,y}]

Slope

Slope[a^2==x^2+2 x^3+5 x^4+6 x^5+6 x^6+4 x^7+x^8+y^2+2 x y^2+2 x^2 y^2+4 x^3 y^2+14 x^4 y^2+12 x^5 y^2+4 x^6 y^2-3 y^4-2 x y^4+10 x^2 y^4+12 x^3 y^4+6 x^4 y^4+2 y^6+4 x y^6+4 x^2 y^6+y^8,{x,y}]//FullSimplify